Calculus: Early Transcendentals (2nd Edition)

by
Briggs, Bill L.; Cochran, Lyle; Gillett, Bernard

Answer

It is meant that both the denominator and the numerator of the expression under the limit approach $0$ as $x$ tends to $a$.
The examples are $$\lim_{x\to0}\frac{\sin x}{x},\quad \lim_{x\to0}\frac{x^3}{2x},\quad \lim_{x\to0}\frac{1-\cos x}{\sin x}.$$

Work Step by Step

It is meant that when $x\to a$ then both the numerator and the denominator of the expression under the limit approach zero. The examples are
1) $$\lim_{x\to 0}\frac{\sin x}{x}$$
because as $x\to0$ the numerator $\sin x$ approaches $0$ (sine is a continuous function and $\sin 0 =0$) and the denominator is just $x$ that already goes to zero.
2) $$\lim_{x\to0}\frac{x^3}{2x}.$$
When $x\to0$ then every positive power of $x$ approaches zero so both $x^3$ and $2x$ tend to zero.
3) $$\lim_{x\to0}\frac{1-\cos x}{\sin x}.$$
We already showed that when $x\to0 $ then $\sin x\to0$. Also when $x\to 0$ then $\cos x\to 1$ because cosine is a continuous fucntion and $\cos 0=1$. Thus $1-\cos x$ tends to zero.